Developing A Linear Equation From A Word Description
Theory
Many real-world problems can be solved by translating the words into a linear equation, then solving it. This page covers the four-step word-to-equation method, common phrase translations, and worked examples on number, age, cost-comparison, and average problems.
Many real-world problems can be solved by translating the words into a linear equation, then solving it. The challenge is choosing the right variable, expressing other quantities in terms of it, and writing the relationship between the two sides.
Look for the key relationship words: "total", "sum", "difference", "equals", "is the same as", "average". These tell you what the two sides of the equation are. Words like "increased by", "less than", "twice", "one-third of" tell you how to build the algebraic expressions on each side.
The most important rule: always answer the actual question at the end. The variable you chose might not be what the question is asking for — re-read the question after solving.
Common phrases and their algebraic translations:
| Words | Algebra |
|---|---|
| "a number" | |
| "twice a number" | |
| "a number, increased by 5" | |
| "5 less than a number" | |
| "one-third of a number" | |
| "the next consecutive number" | |
| " |
Common relationship patterns:
The four-step method
- Let
- Write each other quantity in terms of
- Form an equation that captures a relationship from the question (often a total, a difference, or "the same as").
- Solve the equation, then check the answer makes sense in the original context — and answer the actual question.
Common problem types. "Find the number" — let
Let
Let
Jerry has
Let
At
Let
Common pitfalls
Frequently asked questions
How do I turn a word problem into an equation?
Let x equal something specific (often the smallest unknown or the quantity the question asks about). Then write every other quantity in terms of x. Finally, write an equation that captures a key relationship from the problem — usually a total, a difference, or something being equal.
What does '5 less than a number' mean as algebra?
It is x minus 5, not 5 minus x. The order matters: 5 less than a number means you start with the number and take 5 away.
How do I solve a 'when are the costs equal' problem?
Write a formula for each cost in terms of the same variable (such as number of texts or minutes). Then set the two formulas equal to each other and solve for that variable.
What goes wrong with consecutive numbers?
Use x and x plus 1 for two consecutive integers, or x, x plus 2, x plus 4 for consecutive even or odd numbers. The key is that the gap is constant. Then form an equation from the given relationship (often a sum or product).
Why do I need to answer the actual question at the end?
Because the variable you chose might not be what the question is asking for. If x equals 7 but the question asks for the larger of two numbers (which is x plus 5), the answer is 12, not 7. Always re-read the question after solving.
How do I handle 'in 4 years time' age problems?
If a person's current age is x, then in 4 years time their age is x plus 4. If they were 3 years old 6 years ago, their current age is x where x minus 6 equals 3, so x equals 9. Build the equation from how old they will be or were, not their current age.
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Practice Questions
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Practice Questions